Dynamics of correlations of Bose and Fermi particles

Abstract

Communicated by M. LachowiczWe discuss the origin of the microscopic description of correlations in quantum many-particle systems obeying FermiDirac and Bose-Einstein statistics. For correlation operators that give the alternative description of the quantum state evolution of Bose and Fermi particles, we deduce the von Neumann hierarchy of nonlinear equations and construct the solution of its initial-value problem in the corresponding spaces of sequences of trace class operators. The links of constructed solution both with the solution of the quantum BBGKY hierarchy and with the nonlinear BBGKY hierarchy for marginal correlation operators are discussed. The solutions of the Cauchy problems of these hierarchies are constructed, in particular for initial data satisfying a chaos property.

Mean Field Asymptotics of Generalized Quantum Kinetic Equation

“…where the bounded operators g n ∈ L(H n ), n ≥ 2, are specified initial correlations [24]. Such initial data is typical for the condensed states of quantum gases, for example, the equilibrium state of the Bose condensate satisfies the weakening of correlation condition with the correlations which characterize the condensed state [25].…”

“…where A 1+n (t) is the (1 + n)th-order cumulant (5) of groups of operators (2) and the operators g 1+n , n ≥ 0, are specified initial correlations. The series (24) converges under the condition that: F 0 1 L 1 (H) < e −10 (1 + e −9 ) −1 . Correspondingly, the marginal functionals of the state are represented by the following expansions:…”

“…, s) into |P| nonempty mutually disjoint subsets Y i ⊂ Y . The marginal correlation functionals G s t, Y | F 1 (t) , s ≥ 2, are represented by the following expansions[24]…”

On quantum kinetic equations of many-particle systems in condensed states

“…Such initial data is typical for the condensed states of quantum gases, for example, the equilibrium state of the Bose condensate satisfies the weakening of correlation condition with the correlations which characterize the condensed state [13]. We remark that nonequilibrium dynamics of correlations was constructed in [14]. The evolution of all possible states is described by the sequence of marginal density operators F (t) = 1, F 1 (t, 1), .…”

Kinetic Equations of Active Soft Matter

“…We note that, according to equality (21), in the mean field limit the marginal correlation functions defined as cluster expansions of marginal distribution functions [30,33,34] namely,…”